In this explainer, we will learn how to use the properties of addition and multiplication on vectors.
We begin by recalling that a vector is a quantity with both a magnitude and a direction. A vector can be represented in a suitable space by a directed line segment with a specific length. This means that we can think of vectors as defining movement, traveling in a given direction for a specified distance.
This idea allows us to add two vectors together; if both vectors can be thought of as movement in a given direction for a specified distance, their sum can be thought of as the combination of both movements combined.
In two dimensions, we can choose a space where we can represent the magnitude and direction in terms of horizontal and vertical change. In this space, vector has a horizontal component of and a vertical component of . We can think of this as a displacement units horizontally and a displacement units vertically.
This means we can add two vectors together by considering their components. Graphically, the sum of two vectors and is the combined displacement. Therefore, we can sketch the terminal point of the first vector as the initial point of the second vector. Then, the sum of the vectors has the initial point of the first vector and the terminal point of the second vector, as shown in the following diagram.
Since vector represents the displacement of both and , it will have a horizontal component equal to the sum of the horizontal components of and and a vertical component equal to the sum of the vertical components of and . This gives us the following.
Theorem: Addition of Vectors in Two Dimensions
For any two vectors in two dimensions and ,
Since the sum of any two vectors in two dimensions is also a two-dimensional vector, we can say that vector addition in two dimensions is closed. This is sometimes referred to as the closure property of vector addition.
This idea extends to higher dimensions; however, in this is explainer, we will only be working in two dimensions.
We can also define scalar multiplication on a vector as scalar multiplication of its components. Graphically, scalar multiplication of a vector by scalar is a dilation of the vector by a factor .
Theorem: Scalar Multiplication of Vectors in Two Dimensions
For any vectors and scalar ,
Letβs see an example of how to use these definitions to answer a question involving a property of vector addition.
Example 1: Commutativity of Vector Addition
Complete the following: .
Answer
We will start by simplifying the left-hand side of the equation. To find the sum of a pair of vectors, we recall that and .
Then,
In our case, and ; therefore,
This is equal to the right-hand side of the given equation, so we will call the missing vector .
We can then simplify the right-hand side of the given equation:
Setting this equal to the right-hand side of the equation gives
For two vectors to be equal, their corresponding components must be equal. Setting the corresponding components to be equal gives us two equations:
We can solve these to see and , so the missing vector is
There is a second method of showing this. We start by adding the vectors on the left-hand side of the equation:
Then, we use the commutative property of addition:
Finally, we can use vector addition:
Hence, the missing vector is .
The second method in the question above can be generalized to any two vectors:
In other words, for any vectors in two dimensions and ,
This is known as the commutativity of vector addition. A graphical interpretation of this property is shown in the following diagram.
If and are nonzero, then we can sketch these vectors as the sides of a parallelogram. Then the vector of the diagonal of this parallelogram can be represented as both and , so these expressions must be equal.
We do need to deal with the case where one, or both, of these vectors is the zero vector. If , then we can show that
Hence,
This is called the additive identity property, since adding the zero vector does not change the vector.
We can also demonstrate properties involving scalar multiplication. For example, for any vector ,
Therefore,
This is called the multiplicative identity property, since multiplying a vector by the scalar 1 does not affect its magnitude or direction.
There are many properties of vector addition and scalar multiplication in two dimensions. We will not prove all of these; however, they can all be derived by considering the components of the vectors.
Theorem: Properties of Vector Addition and Scalar Multiplication in Two Dimensions
For any vectors , , and and scalars and , consider the following.
- Properties of vector addition:
- Properties of scalar multiplication of vectors:
All of these properties are true for vectors in dimensions higher than two and provable algebraically. Letβs now see an example of how we can use these properties to evaluate an expression involving vectors.
Example 2: Simplifying a Vector Expression Using the Properties of Vector Operations
Given that and , find .
Answer
We can answer this question directly using the properties of vector addition. First, we will use the commutative property of vector addition to reorder the expression. This says that for any vectors and ,
Applying this to our expression gives
Next, we will use the additive inverse property of vector addition to simplify the expression. This tells us that for any vector ,
Applying this to our expression along with the associative property of vector addition gives
Finally, we will use the additive identity property that says that for any vector ,
Hence,
A second method would be to work with the components of and :
We distribute the negative over the vector by multiplying all of its components by :
Now, we find the sum of the vectors by adding their corresponding components together:
In our next example, we will see a demonstration of how to apply the associative property of vector addition. The method of adding these vectors together by finding the sum of their corresponding components can be generalized to show that the associativity property is true for arbitrary vectors.
Example 3: Checking the Associativity of Vector Addition in Two Dimensions
Consider that , , and .
- Find .
- Find .
- Does equal ?
Answer
Part 1
To find the sum of these vectors, we add the corresponding components:
Evaluating the expression inside the parentheses yields
Adding the corresponding components of these vectors gives
Part 2
To begin,
Evaluating the expression inside the parentheses yields
Adding the corresponding components of these vectors gives
Part 3
We have shown that both of these expressions simplify to give the same vector: . This is an example of the associative property of vector addition. We can use this example to generalize this property.
Let , , and .
Then,
We can then use the associative property of addition to rewrite this vector:
Hence, for any vectors in two dimensions , , and ,
In our next example, we will describe a property of scalar multiplication and prove that the property holds for arbitrary vectors and an arbitrary scalar.
Example 4: Describing a Property of Scalar Multiplication
What is the property that shows that ?
Answer
This property is called the distributive property of scalar multiplication over vector addition. It states that for any vectors and and scalar ,
We can prove this by considering the components of and :
Then, to multiply the vector by scalar , we multiply each component by , giving us
Next, we know that multiplication is distributive over addition:
Finally, we can rewrite this as
This property is known as the distributive property of scalar multiplication over scalar addition.
In the example that follows, we will use the properties of vectors to help us determine a missing vector from a vector equation.
Example 5: Checking the Distributive Property of Scalar Multiplication over Vector Addition
Complete the following: .
Answer
We start by simplifying the left-hand side of the equation. First, we use the fact that scalar multiplication is distributive over vector addition:
We can then evaluate the scalar multiplication:
Equating this with the left-hand side of the equation gives
We can then simplify this equation using the elimination property of vector addition that tells us that if , then .
To make this clear, we will use the commutative property of vector addition to rewrite our equation as
Then, we eliminate the vector , giving us
Hence, the missing vector is .
In our final example, we will prove the additive inverse property of vector addition.
Example 6: Describing the Property of Additive Inverses for Vector Addition
What is the addition property that shows that ?
Answer
This property is called the additive inverse property of vector addition. We can prove this property by considering the components of vector . First, let .
Then,
We can then substitute this into our expression and evaluate:
This additive inverse property of vector addition states for any vector ,
Letβs finish by recapping some of the important points of this explainer.
Key Points
- We can use the properties of vector addition and scalar multiplication to simplify expressions involving vectors.
- Properties of vector addition:
- Properties of scalar multiplication of vectors:
- We can prove that these properties hold by considering the components of the vectors.
- Although we have only considered these properties for vectors in two dimensions, all of these properties extend to vectors in higher dimensions.
